Introduction

Surface acoustic waves (SAWs) have gradually become an essential part of modern civilization since interdigital transducers (IDTs) were invented in 19651. Using IDTs to generate, receive and control on-chip SAWs in wavelengths ranging from 10−8 to 10−4 m, the unique advantages of SAWs can be well applied to today’s microwave electronics in wireless communicating, sensing, navigating2,3,4,5. Compared with electromagnetic waves, SAWs of the same frequency have a much shorter wavelength (up to 10−5), which dramatically increases the possibility of device miniaturization. Also, as quasi-two-dimensional waves on solids, the transmission mode of SAWs is relatively simple, easy to distinguish, and low loss. Today, SAW technology is rapidly develo** in the interdisciplinary fields of photonics, biomedical, and quantum science, giving birth to a significant number of interests in, e.g., acousto-optic modulations6,7,8, phonon-based quantum information9,10,11,12,13, spin–phonon interactions14,15, and acoustic microfluidics16,17.

All of the above studies are expected to obtain more precise SAWs manipulation, making high-quality SAW waveguide a key research front. Recently, several kinds of chip-scale SAW waveguides have been put forward, such as those based on guiding modes of phononic crystals18,19, suspended beam structures20,21, and ridges structures obeying impedance mismatch principle14,22. These cases successfully demonstrated the effective operation of SAWs accompanied by rich acoustoelectric and acousto-optic coupling effects. However, there are two fundamental challenges in this direction, i.e., (1) how to eliminate the backscattering that readily occurs when the SAWs detour quickly or passes through intersections, (2) how to match efficiently broadband SAW waveguides with broadband IDTs without causing more insertion loss and reducing the signal-to-noise ratio of the whole electro-acoustic devices. With the rise of research on topological effects in condensed matters, analogous quantum Hall families23,24,25,26,27,28,29,30 have currently been adapted to bosonic systems, e.g., photonics31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47, mechanics48,49,50,51,52,53,54,55,56,57,58,59,1a–c, there are two inequivalent phononic ‘atoms’ (micro-resonator pillars) in each unit cell, marked as pillar A and pillar B, respectively. These two sets of pillars have the same height (h = 8 μm) and side angle (θ = 6°) but may differ in their radius. We define a relative mass Δm for all these A and B pillars and Δm\(=\frac{{r}_{A}-{r}_{B}}{{r}_{A}+{r}_{B}}\) for their contrast. When Δm = 0 (e.g., rA = rB = 5.8 μm in Fig. 1e), a “semimetal” for SAWs could be formed80, the symmetry of the honeycomb crystal preserves a pair of SAW Dirac degeneracy around the \(\bar{{{\mbox{K}}}}\) and \(\bar{{{\mbox{K}}}}\)’ points of the 1st Brillouin zone (considering the influence of the anisotropic LiNbO3 substrate, \(\bar{{{\mbox{K}}}}\)/\(\bar{{{\mbox{K}}}}\)’ are used here to distinguish them from the high symmetry points K/K’ of the traditional honeycomb lattice. See details in SI Note 7). When Δm ≠ 0 (e.g., rA = 5.9 μm rB = 3.9 μm in Fig. 1d; and rA = 3.9 μm rB = 5.9 μm in Fig. 1f), the spatial inversion symmetry (along the y-axis) of the crystal is then broken, thus leading to “insulators” for SAWs with omnidirectional bandgaps.

Fig. 1: Our on-chip phononic crystals for SAWs.
figure 1

a Schematic of our phononic crystals containing two sets of micro-resonator pillars, marked as A and B, on a y-cut LiNbO3 semi-infinite half-space. b Unit cell of our phononic crystal, its lattice constant a = \(42/\sqrt{3}\) μm. c schematic of two inequitable pillars. d–f (upper) SEM images and (bottom) calculated band structures of three different phononic crystals with different topological phases. e A SAW Dirac “semimetal” with the same A and B pillars (rA = rB = 5.8 μm). d and f Two SAW valley “insulators” with inequivalent A/B pillars in their radius, both having omnidirectional bandgaps at their \(\bar{{{\mbox{K}}}}\) and \(\bar{{{\mbox{K}}}}\)’ points of the 1st Brillouin zones, i.e., the valleys. We call the “insulator” in d (rA = 5.9 μm rB = 3.9 μm) as valley insulator A (VA) and the “insulator” in f (rA = 3.9 μm rB = 5.9 μm) as valley insulator B (VB). Insets show the SAW flux vortices observed at their valleys.

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In these SAW insulators, flux vortices of the SAWs could be observed at their frequency extrema \(\bar{{{\mbox{K}}}}\) (\(\bar{{{\mbox{K}}}}\)’) points [insets of Fig. 1, and Fig.S9 of Supplementary Information (SI)], which is the characteristic of the classic wave analogous to the electronic valley states73. By constructing zigzag boundary between different SAW insulators in Fig. 1d and f, we realized the analogous QVHE for SAWs, or say SAW topological valley-locked states (TVSs). Simulation and experimental results are shown in Fig. S10 of SI. Note that the symmetry of the y-cut LiNbO3 substrate slightly affects our phononic crystal’s symmetry. However, this effect is faint and does not affect the presence of TVSs (see SI Note 7).

Extended topological valley-locked states for SAWs

Waveguides built from topological edge states may provide a disruptive advantage over conventionally designed ones, e.g., backscattering suppression, single-mode operation, and linear dispersion at their working bandwidth. It is especially the case for SAWs. Due to the relatively low impedance mismatch (generally <101) in different solid-state materials, and the extensively high sensitivity of acoustic waves in solid-state media, conventional designed solid-state acoustic waveguiding are pretty susceptible to intersections, sharp bends, and wavelength-scale defects.

Meanwhile, for (future) electro-acoustic SAW devices with SAW waveguides functioned between dual- (or multiple-) port IDTs, there is a crucial trade-off between the signal fidelity and the working bandwidth17. Broadband operating frequency is one of the core requirements of advanced signal processing. Generally, a broadband IDT requires a relatively large “aperture/wavelength” ratio, which is bound to reduce the proportion of the SAWs injected into the waveguide to the total SAWs that the IDT pumped. For the device as a whole, this requirement would increase the insertion loss of the waveguide signal and reduce its fidelity (see more in SI, Note 10). Consequently, to the practical use of SAW waveguides in concreting, e.g., phononic integrated circuits, a topological SAW waveguide with a wide aperture that can match broadband IDTs, is highly promising.

In this work, our second step (and the main achievement) is to realize such wide-aperture topological waveguides for SAWs by bringing about the ETVSs. Specifically, as shown in the upper panel of Fig. 2, based on the valley-type phononic crystal present in Section I, we doped our SAW semimetal (Δm = 0, rA = rB = 5.8 μm) in the interface between two SAW insulators with opposite valleys [i.e., one Δm = 0.2 (rA = 5.9 μm rB = 3.9 μm) and one Δm = −0.2 (rA = 3.9 μm rB = 5.9 μm)], thus directly making a heterostructured interface. This do** could extend the one-dimensional (1D) TVSs to quasi-two-dimensional (quasi-2D) ETVSs, with an additional degree of freedom (DOF) in the width of the do** area. Topological characteristics carried by the bulk on both sides of the heterostructured interface will tunnel through the do** area with Dirac dispersion83,84, thus forming the ETVSs.

Fig. 2: Extended topological valley-locked states for SAWs.
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a–c (bottom) Schematics of three heterostructured valley interfaces (see SEM images in Fig. S2), with increasing molecular layers (from 3, 5, to 7, respectively) of SAW “semimetal” (SM) doped inside a VA and a VB. (upper) Experimentally measured SAW out-of-plane amplitude distributions crossing the do** areas. d–f Calculated and experimentally measured band structures of the heterostructured interfaces. The color bar represents the density of energy. g–i Simulated and experimentally imaged SAW out-of-plane displacement and phase distributions of the heterostructured interfaces (at 76 MHz shown in the figures), demonstrating a pseudo-diffusion-like phase uniformity of these edge states.

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We numerically modeled several of those heterostructured interfaces with increasing do** areas. Projected band structures of these heterostructured interfaces and their SAW field/phase distributions are shown in the middle and bottom panels of Fig. 2. All dispersions of these ETVSs share the same characteristics as the conventional TVSs. Differently, for the TVSs, their energy is highly localized around the 1D interfaces. For the ETVSs, their energy fills the entire 2D do** area, thus offering a topological waveguide with a much wider aperture. As the do** area increases, the waveguide aperture may expand several times or even dozens of times to the original (none doped) one, at the price of a gradual decrease in working bandwidth. (see SI, Note 8).

Experimentally, we fabricated three different heterostructured samples with 3, 5, and 7 molecular do** layers. All the samples are in the same configuration, i.e., phononic crystal containing the heterostructured interface is prepared in the middle; two broadband IDTs, acting as SAW emitter and SAW receiver, are prepared on both sides, respectively. In the experiments, the emitter IDTs pump planar SAWs into the heterostructured interfaces (i.e., the waveguides), exciting the ETVSs. A laser vibrometer imaged the out-of-plane displacement of the excited ETVSs, confirming the SAWs do strongly localized in the whole do** area, together with uniform phases. After the SAWs passed through the heterostructured interfaces, they were captured by the receiver IDTs and read out by a network analyzer, demonstrating considerable transmittance (i.e., S21, Fig. S18d, f, h of SI) at their operating frequencies. For the 3, 5, and 7-layer do** ETVSs, their maximum working bandwidths are expected to reach 7.8%, 6.7%, and 5.3%, respectively, without additional challenge for sample processing (see SI, Note 9).

ETVS based splitter with high SAW throughput

A three-port SAW splitter sample was fabricated to confirm that the ETVSs hold the same topological properties as TVSs. As shown in Fig. 3, this sample consists of five different parts, thus forming four ports labeled in the figure. Ports #1 and #3 are located on the straight SAW transmission route but are different in their valley pseudospins. Ports #2 and #4 are located on two “zigzag” SAW transmission routes with 120° sharp bends but hold the same valley pseudospin as Port#1. Planer SAWs wave generated by IDTs and are injected into the multi-port device through Port#1.

Fig. 3: High throughput SAW splitter based on the ETVS.
figure 3

(bottom) Schematic of a three-ports SAW splitter sample consists of the (blue shaded) VA, (red shaded) VB, and (unshaded) SM. In the experiment, planar SAWs are injected into the sample from Port#1 and captured from the other three ports. (upper) Calculated SAW out-of-plane displacement distributions and experimentally imaged SAW amplitude around all four ports. All SAWs are highly localized in routes that hold the same valley pseudospin, i.e., the SAWs are output the sample only via Ports #2 and #3.

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In conventional SAW waveguide devices, the input SAWs will mainly directly exit the device via Port #4 but partially go through the “zigzag” routes to reach Ports #2 and #3. However, in this topological valley-locked waveguide, the input SAWs from Port #1 could only exit the system via the ports that hold the same valley-pseudospin. Hence, these SAWs should pass through those “zigzag” routes and eventually be captured at Ports #2 and #3. Experimental imaged SAW distributions around the input and output ports confirm that the SAWs experience this “Port#1 → Ports# 2/3” transport. These results agree well with theoretical expectations, proving that in our SAW system, the ETVSs do hold the same function as conventional TVSs. Our SAW splitter is a demo for the valley-pseudospin locking and a promising primary component for future SAW integrated circuits with high throughput and energy capacity.

Defects immunity and phase reconstruction for SAWs

In the QHEs and QSHEs35,

Fig. 4: Comparison of wavelength-scale defect robustness between the TVS and ETVS for SAWs.
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(Left panels) the TVS case; (right panels) the ETVS case. a, b (bottom) Schematics of a TVS waveguide and an ETVS waveguide have two uniform wavelength-scale vacuum defects embedded. Blue, red, light blue shaded areas are the VA, VB, and the defects; the SM remains unshaded. In the experiment, the same planar SAWs are injected into these two samples. (upper) Calculated SAW out-of-plane distributions on the surface of the two samples. 1D red and blue bars are experimental measured SAW out-of-plane amplitude before and behind the defects. c, d Calculated and experimental imaged SAW out-of-plane displacement distributions before and behind the defects in both the waveguides.

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Unlike non-doped TVSs in conventional QVHE systems, ETVSs have an excellent tolerance for the same wavelength scale defects. The additional width (DOF) of ETVSs endows them with stronger robustness in their transmission routes since the same defect should impose much less disturbance on quasi-2D ETVSs than 1D TVSs. This robustness was observed in a comparative experiment, as shown in the right panel of Fig. 4. When time-harmonic SAWs are pumped into a doped edge, much more SAWs can now be detected from the exit. Along the transmission route, the phase of the SAWs is disturbed around the defect. However, the phase uniformity would quickly emerge as the SAWs propagate away from the defect for several wavelengths. This intriguing phenomenon would bring us some novel SAW applications with precise wavefront control, e.g., SAW cloaking, focusing, collimating, and beam expanding.

ETVS’s stability and configurability

The existence of the topological edge states is originated and preserved by the nontrivial topology of the bulk bands. Although the characteristics and functionalities of those edge states, e.g., defects immunity and insensitivity of the external field perturbation, have been widely demonstrated in electronics, photonics, and phononics, do** (and manipulating) on these topological edges are currently underdeveloped and reveals a promising research area to explore complex nontrivial physics and applications86 in, e.g., signal processing and information communications.

For our present SAW ETVSs, we will demonstrate their stability and configurability to the different do**. Our experiments are shown in Fig. 5. The do** width is adjustable by adopting different do** layers, i.e., layers of the SAW semimetal embedded in the do** area. As the do** width increase, the frequency range in which the ETVSs exist would continue to narrow and eventually disappear. Except that, in a fixed do** width, the do** “height” is also adjustable. This “height” can be defined as the Dirac frequency of the SAW semimetal. By changing the radius of the micropillars, the do** “height” changes from 77.90 MHz, 75.99 MHz, 74.94 MHz (Fig. 5a, b, and c, respectively). Calculated projected band structures of these different heterostructured interfaces (see Fig. S20 of SI) show that the ETVSs crossing the whole bulk band stably exist in all these cases, except for slight frequency shift.

Fig. 5: ETVS’s stability to the do** of different energy barriers.
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a–c (bottom) Schematics of three heterostructured valley interfaces, when the radius of the micropillars of SMs increasing from a 4.3 μm to b 5 μm to c 5.8 μm. (upper) SAW band structures of the corresponding SMs. Their SAW Dirac frequency decreases inside the VA/VB’s bulk bandgap. (middle) Experimentally measured SAW out-of-plane amplitude distributions crossing the do** areas. SAWs remain highly localized in all these cases. d–f Experiment and simulation SAW out-of-plane displacement distribution (at 76 MHz shown in the figures) depending on different do**.

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Experimentally, these configurable ETVSs are imaged (see Fig. 5e–g), all exhibit phase uniformity with slightly changed patterns. When the do** “height” is relatively high, the interaction between adjacent micropillars seems to be relatively weak; SAWs in the do** area (i.e., the ETVSs) are more similar to plane waves. As the do** “height” decreases, the inter-pillar coupling becomes stronger; SAWs in the do** area are more localized around the micropillars, exhibiting a “fish scales” style. Notably, the transmittances of the three ETVS waveguides are unity since they are all valley pseudospin locked and robust to wavelength-scale defects and interactions. Practically, it provides a new strategy to manipulate SAWs in the ETVS waveguide, e.g., operation frequencies and distributions84.